Proof of a Spectral Property related to the singularity formation for the L2 critical nonlinear Schrödinger equation

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Blow up for the critical generalized Korteweg–de Vries equation. I: Dynamics near the soliton (2014, methodology): “See also Merle, Raphaël and Rodnianski [32] for the treatment of the Schrödinger map system and Raphaël and Schweyer [40] for the parabolic harmonic heat flow.”
On Blow-up Solutions to the 3D Cubic Nonlinear Schrödinger Equation (2010, result): “Rapha¨el’s result [21] draws upon a large body of breakthrough work by Merle-Rapha¨el [12]-[17] [21] (see also Fibich-Merle-Rapha¨el [ 6 ]) on the blow-up problem for the L2 critical NLS.”
Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS (2010, methodology): “Let us however insist onto the fact that a very specific algebra is at hand here which relies on the critical mass assumption and avoids both the standard troubles when running two dimensional Morawetz estimates for Schrödinger, and the delicate study of explicit quadratic forms like in [20], [26], ”
Existence and Stability of the log–log Blow-up Dynamics for the L2-Critical Nonlinear Schrödinger Equation in a Domain (2007, background): “Recently, a proof which involves numerical computations was derived in [10] for dimensions N = 2, 3, 4, which covers the physically relevant case N = 2.”